parent narrative

The everyday world for a young child is full of opportunities to engage with number and quantity. From the first few days of life, infants pay special attention to number in their environments. Babies’ everyday experiences with quantity provide the foundation for more advanced math concepts that develop throughout early childhood and beyond. Young children are actually able to think about math in remarkable ways as they get older. However, their thinking about number and quantity becomes increasingly dependent on the support they receive from parents and caregivers.

Young children benefit from mathematical experiences that are not only fun and challenging, but also from activities that fit with their intuitive ideas about number. In fact, the more math activities and conversations that fill their day, the more prepared children, especially toddlers, will be when they start school. For these reasons, it is important for caregivers to be aware of the mathematical potential of the children in their charge. Caregivers can use information about how children think about number and quantity to create environments that will maximize their mathematical potential.

Numerical Aptitude (0 – 12 Months)

In the first year of life, infants think about number in simple ways. Babies have the ability to see whether two small quantities are different or the same. For example, six-month-old Jessica can tell that a picture of two balls is different from a picture of three balls. It is important to note that even though this ability is remarkable, infants at this age still do not know which of the two quantities is bigger or smaller. Rather, infants simply recognize that two small amounts are different or the same. Babies can also notice the difference between moving quantities. For example, Jessica can also tell that a puppet that jumps twice is somehow different from the same puppet jumping three times. One conclusion that can be drawn from these examples is that even babies are able to immediately see “twoness” and “threeness” without being able to count yet.

Later in the first year of life, infants’ ability to see the difference between quantities gets better. In fact, well before they turn one, infants can see the difference between small and large sets of objects, but only if one set is at least twice as big as the other. For example, at six months of age, baby Erica could tell that a picture of three blocks was different from a picture of two blocks, but she could also tell that a picture of eight blocks was different from a picture of sixteen blocks. This ability is an important milestone in Erica’s development because she is also able to recognize something important about larger groups of objects.

Between nine and twelve months of age, still before they can count, infants’ ability to discriminate between two larger sets of things gets even better. Now, infants can tell the difference between two large sets of objects even if the sets are closer in size. For example, at ten months of age, Erica can now tell there is a difference between a picture of eight blocks and a picture of twelve blocks. She can also see that a group of six dolls is different from a group of eight dolls. Even though Erica’s ability to tell the difference between two sets of objects has improved, she sees that the sets contain different quantities, but she cannot tell which set is larger or smaller.

This early ability to recognize quantity is not limited to groups of individual objects, such as a certain number of blocks or dolls. Before their first birthday, infants also notice the difference between continuous quantities, such as juice. Erica, for example, can tell that a glass that is half full of juice is somehow different from a glass that is three-quarters full of juice. At the same time, she can tell that two full glasses of juice have the same amount of liquid in them.

Numerical Aptitude (13 – 36 Months)

As infants become toddlers, their understanding of number and quantity continues to improve. It is at this time that children learn how to count. These emerging counting skills make it possible for young children to start thinking about more advanced mathematical operations, such as simple addition, subtraction, multiplication, and division. Indeed, the basis for the mathematical skills that children will need in school begins to develop at a very young age. Caregivers can play a huge role in nurturing toddlers’ beginning math skills by encouraging them to count objects and to compare quantities in their everyday environments.

Making Comparison Judgments

Around fourteen to eighteen months of age, children begin to go beyond simply making “same” and “different” judgments. The average fourteen-month-old can identify not only that two groups of objects are different, but now can also tell which of two groups contains more. However, children do not use counting to make these judgments. Instead, they make simple comparisons based on which set “looks like” it contains more. While this development is impressive, it is important to note that toddlers at this age cannot tell “how many” or by how much one quantity is bigger than the other. Consider fifteen-month-old Ashley. With practice and encouragement, Ashley can consistently tell which of two groups of toys is bigger. For example, she can see that three toy cows is more than two cows, but also that six toy chickens is more than three toy chickens, and that twelve toy farm animals is more than eight toy animals even though she cannot count.

Counting and Arithmetic

The foundation for arithmetic begins very early in life. On average, infants start to show awareness of simple arithmetic at five months of age. At this age, most infants realize that something is wrong when one object added to one other object results in the wrong number of objects. For example, consider six-month-old Ann, who sees two balls hidden by a screen. Then, someone places a third ball behind the screen so that Ann sees the addition. Of course, upon removal of the screen, one would expect three balls to be behind the screen. By using some “trickery,” however, there are only two balls in front of Ann when the screen is taken away. She finds this bizarre and looks a long time at the strange result. This response shows that even babies have a natural sense of what will happen when one object is added to a small set.

By the time children approach their second birthday, they become even more aware of the effects of adding a small quantity to or taking a small quantity away from a group of objects. For example, at two years of age, Joey expects that when two of his three toy trucks are taken out of his toy box, one toy truck should remain in the box. Furthermore, Joey knows that if he already had two teddy bears, and he receives a new teddy bear from his grandmother for his birthday, he now has three teddy bears.

Around two years of age, children begin to count. More specifically, children generally learn their first number words, usually “one” and “two.” At Joey’s second birthday party, Joey could answer the question “How old are you today, Joey?” with “two.” Children first learn the counting sequence by imitating the counting words they hear adults use, but they do not necessarily understand what the number words mean. At the same time, however, children know that number words are important. For example, children between one and two years of age start labeling their toys with number words. While playing with his toy cars, Joey will label his cars with the number words he knows, although not necessarily in the correct order or to express the number of toys he has in all.

Very soon, Joey uses more number words when he counts, even though he leaves out some numbers in the sequence. When asked to count, Joey says “one, two, five, seven.” Although not correct, he seems to know that number words must always be said in the same order. By the time children are close to three years old, their ability to count objects improves. For example, close to Brian’s third birthday, he learns to count the toys in his drawer and the steps that lead to his room. Like Joey, Brian always uses number words in the same order. When he counts his toys, he says, “One, two, three, five, eight.” When Brian counts the steps, he says, “One, two, three, five, eight.” This example shows that children at this age do not yet understand that number words indicate “how many,” but at the same time, they do realize that number words are important and that number words should always be said in the same order. Indeed, children’s counting skills develop quickly!

Children as young as two-and-a-half can also point to each object only once when counting things. They also label each item they are counting with only one number word, even though they may leave some numbers out or put some numbers where they do not belong. For example, at 32 months of age, Maria regularly counts her toy ponies while playing with them. She has four toy ponies. This time, she pointed at the pink pony and said “one,” then she pointed at the yellow pony and said “three,” then she pointed at her blue pony and said “two,” and finally she pointed at her favourite pony, the green one, and said “ten.” Obviously her counting skills need to be developed further before she can accurately count the number of objects in a set, but one important aspect of counting is already present. Maria did not count the same pony more than once. With practice, children between two and three years of age can learn to recite the number words correctly from 1 to 10 and many can accurately count sets of one to four objects, such as three candies or four dolls.

Children at this age can name the next number word (for numbers below 10), although most often they need a bit of help. For example, when Victoria was two-and-a-half,,, her mother asked her, “What number comes right after 5?” Victoria cannot answer the question unless she is given a running start. So her mother says, “one, two, three, four, five…?” Then Victoria is able to say, “six!” This example shows that at this age, children usually think of the number sequence as a “singsong” that cannot be broken up at any point, similar to when they learn to sing the alphabet song. To children at this age, numbers are words without mathematical meaning and are not connected to quantity.

Improved Thinking about Counting and Quantity (37 – 48 Months)

Between the ages of three and four years, however, children begin to understand that numbers can indicate quantity (that is, that the number “5” refers to a set with five things in it). More specifically, four-year-old Sophia knows that when she counts her dolls, the last number she says is the number of dolls she has. For instance, Sophia does not need to recount her dolls if someone asks her how many she has; she can just reply, “I have three dolls” because she now knows that counting can tell her “how many.”

Around four years of age, children also learn that numbers said later in the counting sequence are larger than numbers that come before them in the sequence. Sophia can explain that seven is bigger than five because when you count, you say, “One, two, three, four, five, six, seven. You say five and then seven, so seven is bigger than five.” This connection between number and quantity is an important milestone in children’s numeracy development. It marks the time when children no longer think of numbers simply as words that are memorized and repeated like a nursery rhyme or the alphabet, but rather as ways to express quantities that can be compared. For example, children can use counting to find out if one of their friends received more candy than they did. Nearing their fourth birthday, children’s ability to recognize numbers improves and they can express quantities in different ways. For example, many are able to recognize one digit numerals (1, 2, 3, 4, 5, 6, 7, 8, and 9) and can represent numbers up to 5 using their fingers.

At the same time, children’s knowledge of the counting words grows. By the time they are four years of age; many young children can count to 20 or 30, and sometimes even higher. They also begin to notice number patterns when counting. For example, four-year-old Ellen recognizes that “twenty-one, twenty-two, twenty-three…” contains a pattern she already knows: “one, two, three…” Furthermore, nearing their fourth birthday, or shortly thereafter, children can name the next number without having to count from 1. Some children at this age also learn to count backwards from 5.

Basic Arithmetic and Fair Sharing

Between the ages of three and four years, children’s basic knowledge of arithmetic also gets refined. At three-and-a-half years of age, some children can use objects to figure out correct answers to simple addition and subtraction problems (such as one plus two more, one plus three more, two take away one, and three take away two) even though they cannot do so in their heads and without the objects. Consider Adam, for instance, who is almost four years of age. His mother placed three bingo chips in front of him. Then she covered the chips with a piece of paper. She then placed her hand under the paper and removed one of the black disks. She showed this disk to Adam and then asked, “Adam, can you make your chips look just like mine?” Adam took two black disks and placed them in front of him. This shows that Adam understands how taking away from a collection affects the overall quantity of the collection. Without the disks, however, Adam would have considerably more difficulty in figuring and solving this problem.

Around age three, children also begin to understand equal sharing, which is the beginning of multiplication and division. For example, three-year-olds are able to equally distribute up to ten objects between two people by repeating “one for me, one for you.” For example, when Jessica was three-and-a-half, she could equally divide up her six toy cats between her two dolls by giving one cat to her doll Sally, then one cat to her doll Jane, one cat to Sally, and so on until there were no more toy cats left. Now closer to her fourth birthday, Jessica also knows how many cats each of her two dolls has by only counting one of the shares. Jessica now recognizes that the shares are equal.

Engaging with Other Types of Quantities

Children’s ability to think about quantities that cannot be counted, such as juice, sand, and play dough, also improves at this age. For example, children know that when more juice is added to an existing amount of juice, then it should look like there is more juice (in other words, the glass will look fuller). Similarly, when playing in the sand box, Sam can recognize which of two piles of sand he added sand. He knows that when he shovels more sand on one of the existing piles, then the pile will get bigger.

Nearing their fourth birthdays, children also can measure length by placing two objects next to each other. For example, Jeremy can compare the length of his book to the length of his shoe by placing the items next to each other. Children’s vocabulary when comparing objects also expands; by the time they are four, and sometimes earlier, children use words such as “taller,” “shorter,” “skinnier,” “fatter,” “wider,” and “longer” when they talk about comparing objects.

Improved Counting and Understanding of Number (49 – 60 Months)

What children can do with numbers between the ages of four and six is truly remarkable. For one, their counting skills improve substantially. For instance, by four-and-a-half years of age, many children can now recite more of the counting sequence, sometimes all the way to 100. By five years of age, children can learn to count up from numbers other than 1. For example, instead of starting from 1, children can start the count at any number. At four-and-a-half, Jenny can start counting up from 8: “8, 9, 10, 11, 12…” Also, most children at four-and-a-half can learn to count backwards from 5; as they get closer to their fifth birthday, they know how to count backwards from 10, and by five-and-a-half, most learn how to count backward from 20. Around their fifth birthday, many children learn to skip count by 10s (e.g., “10, 20, 30, 40…”).

Also, most four-year old children can count about 9 objects without error. At five years of age, most can count a set of 20 objects accurately and by the time most children are six, they can count 28 objects, and sometimes more, without miscounting. Children also learn to use quicker counting strategies, such as pattern recognition. For example, many children at this age are able to immediately recognize that the pattern represents 6, presumably from their experiences with dice and dominoes. Children who are five and six can learn to use basic arithmetic together with their ability to recognize patterns. For example, five-and-a-half year old Suzie was asked how many dots were in the display

? She knew there were 7 dots, and arrived at this answer by adding 1 to the 6 she immediately saw in the domino pattern.

Representing Number

It is also during this time (49 – 60 months) that children’s ability to recognize written numbers and to communicate about quantity improves. For example, around five years of age, they learn to use finger patterns to represent numbers up to 10 and they can read and write one-digit numerals. As they approach their sixth birthday, children can write numbers in the teens; by six years of age, they can learn to write all two-digit numbers. Finally, around five-and-a-half or six years of age, they can identify written number words (“one,” “two,” etc.) and connect these words with the numbers they correspond to (1, 2, etc.) as well as the quantities they represent.

Improved Understanding of Counting

During the 49 to 60 month period, children learn more about how counting helps us think about number and quantity. When they are given, for example, twelve raisins and told that there are twelve raisins, they realize that someone must have counted these raisins to find how many there were altogether. Consequently, when asked to place twelve raisins in a box, a child at this age would simply put all of the raisins in the box rather than count them out again. What this ability shows is a growing understanding that numbers can be used both as counting words and as words that represent the total number of items being counted. This understanding is a key development in numeracy, because having this knowledge allows children to use “counting on” to solve simple addition problems. That is, when solving the problem “I have five dinosaurs and my sister gave me three more. How many dinosaurs do I have now?” five-year-old Joey is able to start the count at 5 and count on 2 more to get a total of 7. Joey explains. “Well, you have 5 dinosaurs to start and then you get 2 more.” As he holds up one finger, Joey says, “so that means 6 is one more and [while holding up a second finger] 7 is two more.” This way of counting saves time when solving addition and subtraction problems.

Furthermore, by the time children are five years of age, they recognize that once a set is counted, the number words used do not adhere to the specific objects that were tagged. Belinda, who is 5 years of age, knows that it does not matter how you start counting: the yellow block can be labeled “one” and the pink block “two” just as easily as the other way around. She knows that the words “one” and “two” do not belong to any blocks in particular.

Problem Solving and Thinking about Bigger Numbers

By the time they are five years of age; most children can solve a variety of simple addition and subtraction word problems, usually with the use of objects, such as blocks or even drawings. In kindergarten, many, if not most children can even solve multiplication and division problems, also with the use of concrete materials to “act out” the objects and actions in the problem. In the middle of her kindergarten year, for example, Miranda could get the answer to the following multiplication problem: “There were 3 plates on the table with 4 cupcakes on each plate. How many cupcakes in all?” She counted out four blocks and put them in a group. Then she counted out another four blocks and placed them in a separate pile. She repeated this once more and then counted all the blocks until she got to twelve.

Miranda was then given the following division problem: “James had 15 tin soldiers and wanted to share them between with his two friends. How many tin soldiers would each kid get?” She proceeded to count out fifteen blocks and deal them out one by one into three piles. She then counted the number of blocks in each pile and announced that each friend would get five soldiers. This process shows that young children have natural abilities to solve multiplication and division problems, even though they do not understand the mathematical symbols that are used in school for such problems (in these examples 3 x 4 = 12 and 15 ÷ 3 = 5). What is important here is that even children around six years of age have an intuitive sense that the basis for multiplication and division is equal groups of items. This sense allows for quick problem solving strategies, such as counting the number of items in one share instead of counting the items in all the shares.

Related to the idea of equal groupings is the concept of groups of ten, the main concept essential to the number system. By giving children problems that involve groups of ten allows children to develop an understanding of the way numbers work, which can then be used to answer more complicated math problems. Children between the ages of five and six are indeed capable of understanding that a bundle of ten popsicle sticks is the same as ten individual sticks, and later in this period, they can learn that a bundle of 18 popsicle sticks is the same as a bundle of ten plus eight individual sticks. Furthermore, division problems with remainders can introduce children to the concept of fractions. For example, when five-year-old Declan is faced with sharing five cookies with his sister, he knows that one of the cookies should be split in two for the sharing to be fair. Between the ages of four and six, most children can recognize “one half” and can learn to correctly label halves using the term “one half.”

Children in this age range also develop quicker and less concrete methods for solving addition and subtraction problems. By the time they are five years of age, some children choose to reject their older “hands-on” methods; it is at this age that children begin to use number facts to solve a variety of simple addition and subtraction problems. Between the ages of five and six, most children know and use doubles facts to 10 (e.g., “2 + 2 = 4,” “3 + 3 = 6”).

Measurement

Finally, children’s thinking about quantities that cannot be counted (such as length, weight, and volume) also improves between the ages of five and six. Five-year olds have a better understanding of length and measurement compared to younger children. For example, around four years of age, children compare the length of objects by putting them side by side, whereas by the time they are five; many children are able to use objects, such as string, to compare the length of objects. For instance, Kevin, who is five, can use a string to find out that the width of the door is larger than the height of his desk, but only if one of these objects is as long as the string. This development in measurement shows a big improvement in children’s thinking about quantities such as length.

Soon after, children are capable of more complex thought about measurement. They can reason that if the size of a book is shorter than a string, and the string is shorter than the plant, then the book is also shorter than the plant. Furthermore, children in this age range also learn to use other non-standard objects when they measure. For example, between the ages of five and six, children can learn to place paperclips end-to-end to measure the length of a book, and later can learn to measure length using more conventional units, such as centimeters or meters.

Environments that Support the Mathematical Development of Young Children

The information presented here indicates that young children are able to think about number and math in very advanced ways. Clearly, however, not all children are the same and as such, they develop in different ways and at different speeds. Most differences between children’s math knowledge is because they are given different opportunities to engage with and talk about numbers in the home and in preschool environments. Therefore, while it is important to know what children can do, it is also important to know what kinds of numeracy activities to provide children as they prepare for the first years of schooling.

A general principle for engaging children in numeracy activities is to start early. Interacting with children even at very young ages about number and quantity prepares them for the types of math activities they will encounter in school. It is therefore very important to engage toddlers (and perhaps even younger children) in activities that serve to highlight the numbers and patterns found in their everyday lives. Asking children to estimate the number of buttons in the sewing box, for example, and comparing the number of buttons to the number of cookies in the jar is one way to connect number and quantity for the young child. Categorizing objects into rows and columns, for example, or handing out snacks to everybody in the room provide other types of activities that can present children with several big mathematical ideas, including multiplication and division. Other activities, such as figuring out how many more plates are needed on the table so that everyone gets one or determining how old Jimmy was two years ago form the foundation for other mathematical concepts.

Counting helps children to think in these ways. In fact, learning number words helps children to make sense of number. Children should be encouraged to count often, using concrete materials that interest them. They should practice counting things around them, such as the steps that lead to the door of their preschool, the kids in the room, or the legos in the toy box. Activities that involve associating number words with patterns, such as those seen on dice or dominoes, are also important because they help children understand that quantities can be displayed in many different ways. In fact, playing with board games and card games that require children to think about number in a variety of ways is particularly effective.

Language development is also an important part of children’s numeracy development. It is important to encourage children to talk about the quantities they are thinking about by using vocabulary related to number, counting, quantities, and comparisons. Using words such as “taller,” “skinnier,” “colder,” “lighter,” and so on, will make it easier for children to make connections between numbers and the quantities they pay attention to in the world around them. Also, talking with children about their thinking is essential because it provides the adult with information about children’s naturally developing ideas about number. This information allows the parent/caregiver or educator to identify the most effective activities for the child’s continuing development of numeracy.